Let me preface this by saying that I'm an idiot and used "irassharu" in that example about trains. Forget it, it's a sonkeigo, I was tired.
Crap, I feared this question would be asked :lol. It's been a long while since I last studied keigo. <looks for his 60-page booklet on keigo>
This could be a bit long and hard to follow.
In a nutshell, おる、申す forgot about this one, but don't ask for any example, it still confuses the hell out of me , 参る and 致すare different from all other kenjougo (e.g. 伺う、申し上げる、お目にかかる、差し上げる
for one fundamental reason: the former are used to express politeness towards the person you're talking to (kinda like teineigo), whereas the latter express modesty towards the beneficiary of the action.
Examples using 伺う (kenjougo I) and 参る (kenjougo II/teichougo):
1)「先生のところに伺います。」
2)「弟のところに伺います。」
3)「先生のところに参ります。」
4)「弟のところに参ります。」
[Note that all these examples assume that the person you're talking to and the beneficiary of the action the sensei or your little brother are two different people.]
1) Possible. It expresses modesty towards your teacher (伺う)+ politeness towards the person you're talking to(ます).
2) Impossible. Because 伺う is a kenjougo, it necessarily expresses modesty towards the beneficiary. Said beneficiary is your little brother, lower member of the family, and thus should not require the use of modesty.
3) Possible. Think of this case as you expressing "double the amount" of politeness. The ます form expresses politeness towards your conversational partner, and on top of that, 参る adds another layer of politeness towards said partner (NOT towards your teacher).
4) Possible. Like in 3), you express twice the politeness towards your interlocutor, not the person you're gonna pay a call to, meaning it doesn't matter that you're talking about your brother instead of your teacher.
In other words, the reason why kenjougo II is sometimes called "teichougo" is that it's got nothing to do with modesty towards a beneficiary of the action, but is simply a higher degree of politeness than the regular ます form (teineigo). Note that for this reason, teichougo cannot normally be used in plain form. In modern day language, you'll likely never hear or see 「参る」used as is in a sentence, but rather「参ります」. If you wanna express politeness, the ます form is the required basis. Teichougo like 参る can only be added afterwards, as a way to reinforce said politeness.
Finally, here's the reason why teichougo is often accepted as/confused as kenjougo (hence the name kenjougo II). There are cases where both teichougo and regular kenjougo (= kenjougo I) can be used instead of the other. That's when the person you're talking to and the beneficiary of the action are one and the same. Example 1 and 3 are interchangeable when you're actually talking to your teacher (as you probably know, it's not uncommon in Japanese to address people by their name or function). In the end, both 伺います (kenjougo I) and 参ります (kenjougo II/teichougo) express the same degree of respect towards the same person.
In the end, both terms are accepted (it even seems that kenjougo II is more common than teichougo), and honestly, use whichever you prefer as long as you understand how this differs from kenjougo I. Personally I use the word "teichougo" because I think it's much less confusing and simply more "right" for a grammar nazi like myself
. The sonkeigo/kenjougo tandem focuses on the relationship between the actor and the person who is subjected to the actor's action, while the teineigo/teichougo couple focuses on the relationship between the speaker and the person who's spoken to. As such, the idea of calling teichougo a type of kenjougo doesn't make much sense beyond that one case where the two become interchangeable.
... I hope that wasn't too confusing. We spent so much time on this in class, and I realize it's a bit of a waste of time, but keigo is so hard even for the Japanese that I feel grammar might actually help wrap your head around the categories and functions of all the types of keigo.
[EDIT] I should mention that I've heard things like 『致す」and「申す」in plain form in actual sentences on occasion, but those were always in anime/drama/movies/games, never in real life. Either it's used because it sounds cool in dialogues, or it's an archaic construction.